Introduction

This vignette of package VisualizeSimon2Stage (CRAN, Github, RPubs) documents the visualization of probabilities and operating characteristics of Simon\’s two-stage Phase II design.

Note to Users

Examples in this vignette require that the search path has

library(VisualizeSimon2Stage)
library(clinfun)
library(flextable)
library(ggplot2)

Terms and Abbreviations

Simon’s Two-Stage Phase II Design

Notations

Simon’s two-stage design tests the one-sided hypothesis H_0: p\leq p_u vs. H_a: p>p_u for a binary response in the following steps.

1. Enroll n_1 subjects.

• Early termination, or frail, if \leq r_1 positive responses are observed.
• Move to next stage if >r_1 response are observed.

2. Enroll an additional (n-n_1) subjects.

• Fail, if \leq r total responses are observed.
• Success, or H_0 rejected, if >r total responses are observed.

In this vignette, we use p_u to denote the unacceptable response rate, and p_a the acceptable response rate. The parameter nomenclature of r_1, n_1, r and n follows that of PASS and of function clinfun::ph2simon().

Types of Simon’s two-stage design include

• 'minimax' (default), minimum total sample size n
• 'optimal', minimum expected total sample size \textrm{E}(n|p=p_u)
• 'n1', minimum Stage-1 sample size n_1
• 'maximax', to use up the user-provided maximum total sample size, i.e., parameter nmax of function clinfun::ph2simon()
• 'all', all types listed above

S3 class 'ph2simon'

Function clinfun::ph2simon() returns an object of S3 class 'ph2simon'. The output is printed in the R console using S3 method dispatch clinfun:::print.ph2simon().

Example below provides the various Simon’s two-stage designs for hypotheses p_u=.2, p_a=.4, with Type-I error rate \alpha=5\% and Type-II error rate \beta=10\%.

(x = ph2simon(pu = .2, pa = .4, ep1 = .05, ep2 = .1)) 

 Simon 2-stage Phase II design 

Unacceptable response rate:  0.2 
Desirable response rate:  0.4 
Error rates: alpha =  0.05 ; beta =  0.1 

           r1 n1  r  n EN(p0) PET(p0)   qLo   qHi
Minimax     5 24 13 45  31.23  0.6559 0.108 1.000
Admissible  4 20 14 49  30.74  0.6296 0.058 0.108
Optimal     4 19 15 54  30.43  0.6733 0.000 0.058

S4 class 'ph2simon4'

Function ph2simon4() converts an S3 object 'ph2simon' to an S4 object 'ph2simon4'. The output is printed in the R console using S4 method dispatch getMethod(f = 'show', signature = 'ph2simon4').

x |> ph2simon4(type = 'all')

 Simon's 2-Stage Design

Unacceptable Response Rate: 20.0%
Desirable Response Rate: 40.0%
Controlled Error Rates: α ≤ 5%, β ≤ 10%
Maximum Sample Size Allowed: 100

        r1 n1  r   n EN(pu) EN(pa) PET(pu) PET(pa)    α     β
minimax  5 24 13  45   31.2   44.2   65.6%    4.0% 4.8% 10.0%
optimal  4 19 15  54   30.4   51.6   67.3%    7.0% 4.8%  9.6%
n1       3 15 19  71   34.7   65.9   64.8%    9.1% 4.5%  9.8%
maximax  7 27 28 100   38.4   93.0   84.4%    9.5% 1.2%  9.9%

Probabilities

Math

Given a Simon’s two-stage design (r_1,n_1,r,n) and a true response rate p, we have

• the number of Stage-1 positive responses X_1 \sim \textrm{binom}(n_1, p).
• the number of Stage-2 positive responses X_2 \sim \textrm{binom}(n-n_1, p).
• X_1 and X_2 are independent.

The probability of early termination is p_{\textrm{frail}} = \textrm{Pr}(X_1 \leq r_1)

The probability of failure to reject H_0 is \sum_{s_1 = r_1+1}^{n_1} \textrm{Pr}(X_1=s_1)\cdot\textrm{Pr}\left(X_2 \leq (r-s_1)\right)

The probability of successfully rejecting H_0 is \sum_{s_1 = r_1+1}^{n_1} \textrm{Pr}(X_1=s_1)\cdot\textrm{Pr}(X_2 > (r-s_1))

The expected sample size is \textrm{E}(n) = p_{\textrm{frail}} \cdot n_1 + (1 - p_{\textrm{frail}}) \cdot n

Numbers

The S3 generic simon_pr() calculates the probabilities of early termination, fail and success at one-or-more response probability p, either from an S3 object 'ph2simon',

x |> simon_pr(prob = c(.2, .3, .4)) |> as_flextable()

or from an S4 object 'ph2simon4',

x |> ph2simon4() |> simon_pr(prob = c(.2, .3, .4)) |> as_flextable()

or from the design parameters r1, n1, r and n. In such case the user must call the S3 method dispatch simon_pr.ph2simon4() explicitly.

simon_pr.ph2simon4(prob = c(.2, .3, .4), r1 = 5L, n1 = 24L, r = 13L, n = 45L) |>
  as_flextable()

Visualization

The S3 generic ggplot2::autoplot() visualize the probabilities of early termination, fail and success at p_u and p_a. The donut slices for success are colored with the highest opacity, indicating that they represent the Type-I error rate \alpha if p=p_u, and the power 1-\beta if p=p_a. The probabilities are visualized either from an S3 object 'ph2simon',

x |> autoplot(type = 'optimal')

or from an S4 object 'ph2simon4'.

x |> ph2simon4(type = 'optimal') |> autoplot()

Simulation

The S3 generic r_simon() simulates the number of positive responses in Simon’s two-stage design. Following examples showing simulations at p_u=.2, either from an S3 object 'ph2simon',

set.seed(15); r_u = x |> r_simon(R = 1e4L, prob = .2, type = 'optimal')

or from an S4 object 'ph2simon4',

set.seed(15); r_u1 = x |> ph2simon4(type = 'optimal') |> r_simon(R = 1e4L, prob = .2)
stopifnot(identical(r_u, r_u1))

or from the design parameters r1, n1, r and n. In such case the user must call the S3 method dispatch r_simon.ph2simon4() explicitly.

set.seed(15); r_u2 = r_simon.ph2simon4(R = 1e4L, prob = .2, r1 = 4L, n1 = 19L, r = 15L, n = 54L)
stopifnot(identical(r_u, r_u2))

It is easy to illustrate that the Type-I error rate is controlled at \alpha<5\% when p=p_u=.2,

r_u |> attr(which = 'dx', exact = TRUE) |> table() |> 
  as_flextable() |> set_caption(caption = 'pu = .2')

and the Type-II error rate is controlled at \beta<10\%, or power (1-\beta)>90\%, when p=p_a=.4.

set.seed(24); x |> r_simon(R = 1e4L, prob = .4) |>
  attr(which = 'dx', exact = TRUE) |> table() |> 
  as_flextable() |> set_caption(caption = 'pa = .4')

Operating Characteristics

Suppose we have three drugs A, B and C, with true response rate p^A=.3, p^B=.2 and p^C=.15, respectively.

p = c(A = .3, B = .2, C = .15)

We simulated 1e4L Simon’s two-stage trials of each drug to obtain the operating characteristics, i.e., the percentage of trials in which each drug

• having the highest number of responses
• both having the highest number of responses and successfully rejecting the null hypothesis.

The operating characteristics are visualized either from an S3 object 'ph2simon',

set.seed(52); x |> simon_oc(prob = p, R = 1e4L, type = 'optimal')

or from an S4 object 'ph2simon4',

set.seed(52); x |> ph2simon4(type = 'optimal') |> simon_oc(prob = p, R = 1e4L)

or from the design parameters r1, n1, r and n. In such case the user must call the S3 method dispatch simon_oc.ph2simon4() explicitly.

set.seed(52); simon_oc.ph2simon4(prob = p, R = 1e4L, r1 = 4L, n1 = 19L, r = 15L, n = 54L)